Actually, let me clarify:
Let R be the space of real numbers. R x R (pronounced "R cross R") is the set of all ordered pairs (x,y) where x and y are both in R.
When you graph a function f(x) = ?, you are looking at every x and finding the unique y value that lies above it. In fact, the ability to graph f(x) like this is what makes it a function!
Another way to describe the graph of a function, however, is the set of ordered pairs (x,y) in R x R such that y = f(x).
The top diagram in this image is the graph of a function, since there is exactly one point in the graph for each x value. Notice that I'm using "graph" in a very specific sense: it refers to the set of blackened points (which in this case form a wavy line).
We can think of the color black as representing "TRUE" and white as representing "FALSE". So for any point (x,y), being in the graph (a.k.a. being black) is the same as it being TRUE that y = f(x). If (x,y) is not in the graph (in other words, it is white), then it is FALSE that y = f(x). So really, the graph of a function is just the set of points where it is TRUE that f(x) = y.
With this understanding of a graph, we can generalize the idea of a function. The middle diagram shows the graph of what's known as a "relation". A point (x,y) is blackened if and only if x ~ y (read, "x related to y"). In this case, each x value has TWO possible y values where x ~ y.
Just as a function can be defined by its graph, a relation can be defined by its graph. Just as you might ask "Does y = f(x)?", you can ask "Is x ~ y?" for any particular ordered pair (x,y). In fact, a function IS a special kind of relation. It's important to note that with most relations, order is importat! So just because x ~ y, doesn't mean y ~ x!
The bottom image (thrown in for good measure), shows that any graph can be a relation. In this case, there are an infinite number of y values such that x ~ y.
Now, we can define the square root to be a relation! We will say that x ~ y if x is "a square root of" y. And by this, we mean that x ~ y if and only if x^2 = y.
This is in fact a function of x, whether x is positive or negative (or imaginary, for that matter!), because for every x, there is exactly one y satisfying x ~ y.
It turns out that you are asking "Is there a y such that -1 is a square root of y?" Well, we just determined that for any x (including x = -1), we can find a unique such y! That is, we are looking for a y such that (-1)^2 = y. That y is just the number 1.
Now, the reason I brought up all this stuff about relations is that while the "is square root of" relation is a function of x, it does NOT have a natural inverse function. To create an inverse function, people just restrict the domain of the "is square root of" function to only non-negative x.
